Quantum precision enhancement is of fundamental importance for the development of advanced metrological optical experiments, such as gravitational wave detection and frequency calibration with atomic clocks. Precision in these experiments is strongly limited by the 1/√N shot noise factor with N being the number of probes (photons, atoms) employed in the experiment. Quantum theory provides tools to overcome the bound by using entangled probes. In an idealized scenario this gives rise to the Heisenberg scaling of precision 1/N. Here we show that when decoherence is taken into account, the maximal possible quantum enhancement in the asymptotic limit of infinite N amounts generically to a constant factor rather than quadratic improvement. We provide efficient and intuitive tools for deriving the bounds based on the geometry of quantum channels and semi-definite programming. We apply these tools to derive bounds for models of decoherence relevant for metrological applications including: depolarization, dephasing, spontaneous emission and photon loss.

Non-classical states of light find applications in enhancing the performance of optical interferometric experiments, with notable example of gravitational wave-detectors. Still, the presence of decoherence hinders significantly the performance of quantum-enhanced protocols. In this review, we summarize the developments of quantum metrology with particular focus on optical interferometry and derive fundamental bounds on achievable quantum-enhanced precision in optical interferometry taking into account the most relevant decoherence processes including: phase diffusion, losses and imperfect interferometric visibility. We introduce all the necessary tools of quantum optics as well as quantum estimation theory required to derive the bounds. We also discuss the practical attainability of the bounds derived and stress in particular that the techniques of quantum-enhanced interferometry which are being implemented in modern gravitational wave detectors are close to the optimal ones.

Parameter estimation is of fundamental importance in areas from atomic spectroscopy and atomic clocks to gravitational wave-detection. Entangled probes provide a significant precision gain over classical strategies in the absence of noise. However, recent results seem to indicate that any small amount of realistic noise restricts the advantage of quantum strategies to an improvement by at most a multiplicative constant. Here we identify a relevant scenario in which one can overcome this restriction and attain super-classical precision scaling even in the presence of uncorrelated noise. We show that precision can be significantly enhanced when the noise is concentrated along some spatial direction, while the Hamiltonian governing the evolution which depends on the parameter to be estimated can be engineered to point along a different direction. In the case of perpendicular orientation, we find super-classical scaling and identify a state which achieves the optimum.Estimation of an unknown parameter is essential across disciplines from atomic spectroscopy and clocks [1][2][3] to gravitational wave-detection [4]. It is typically achieved by letting a probe, e.g. light, interact with the system under investigation, picking up information about the desired parameter. As seen in Fig. 1, a metrology protocol can be understood in four main steps [5,6]: i) preparation of the probe, ii) interaction with the system, iii) readout of the probe, and iv) construction of an estimate of the unknown parameter from the results. Steps (i)-(iii) may be repeated many times before the final construction of the estimate. FIG. 1.General metrology protocol where a known probe state evolves according to a physical evolution depending on an unknown parameter ω. After sufficient amount of data is collected an estimate for the parameter is constructed.The estimate uncertainty will depend on the available resources, here the probe size N and the total time T available for the experiment (other choices are possible [7]). By the central limit theorem, for N uncorrelated particles, the best uncertainty scales as 1/ √ νN , where ν = T /t is the number of evolve-and-measure rounds. This bound is known as the shot-noise or standard quantum limit (SQL). By making use of quantum phenomena, a metrology protocol may surpass the SQL, reaching instead the limits imposed by the quantum uncertainty relations. For probes of non-interacting particles, the best possible scaling compatible with these relations is 1/( √ νN ), known as the Heisenberg limit. Without noise, the Heisenberg limit can be attained using entangled input states, e.g. Greenberger-HorneZeilinger (GHZ) states for atomic spectroscopy [8]. In the presence of noise however, the picture is much less clear, as the optimal strategy depends strongly on the model of decoherence considered. Nevertheless, the SQL has been significantly surpassed in experiments of optical magnetometry [9,10], which proved that some sources of noise can be effectively counterbalanced [11,12]. However, unless one can k...

Quantum metrology offers enhanced performance in experiments on topics such as gravitational wave-detection, magnetometry or atomic clock frequency calibration. The enhancement, however, requires a delicate tuning of relevant quantum features, such as entanglement or squeezing. For any practical application, the inevitable impact of decoherence needs to be taken into account in order to correctly quantify the ultimate attainable gain in precision. We compare the applicability and the effectiveness of various methods of calculating the ultimate precision bounds resulting from the presence of decoherence. This allows us to place a number of seemingly unrelated concepts into a common framework and arrive at an explicit hierarchy of quantum metrological methods in terms of the tightness of the bounds they provide. In particular, we show a way to extend the techniques originally proposed in Demkowicz-Dobrzański et al (2012 Nature Commun. 3 1063, so that they can be efficiently applied not only in the asymptotic but also in the finite number of particles regime. As a result, we obtain a simple and direct method, yielding bounds that interpolate between the quantum enhanced scaling characteristic for a small number of particles and the asymptotic regime, where quantum enhancement amounts to a constant factor improvement. Methods are applied to numerous models, including noisy phase and frequency estimation, as well as the estimation of the decoherence strength itself.

Quantum metrology protocols allow to surpass precision limits typical to classical statistics. However, in recent years, no-go theorems have been formulated, which state that typical forms of uncorrelated noise can constrain the quantum enhancement to a constant factor, and thus bound the error to the standard asymptotic scaling. In particular, that is the case of time-homogeneous (Lindbladian) dephasing and, more generally, all semigroup dynamics that include phase covariant terms, which commute with the system Hamiltonian. We show that the standard scaling can be surpassed when the dynamics is no longer ruled by a semigroup and becomes time-inhomogeneous. In this case, the ultimate precision is determined by the system short-time behaviour, which when exhibiting the natural Zeno regime leads to a non-standard asymptotic resolution. In particular, we demonstrate that the relevant noise feature dictating the precision is the violation of the semigroup property at short timescales, while non-Markovianity does not play any specific role.Introduction.-Parameter estimation, ranging from the precise determination of atomic transition frequencies to external magnetic field strengths, is a central task in modern physics [1][2][3][4][5][6][7]. Quantum probes made up of N entangled particles can attain the so-called Heisenberg limit (HL), where the estimation mean squared error (MSE) scales as ∼ 1/N 2 , as compared with the standard quantum limit (SQL) ∼ 1/N of classical statistics [8].Heisenberg resolution relies on the unitarity of the time evolution. In realistic situations, however, quantum probes decohere as a result of the unavoidable interaction with the surrounding environment [9]. Such interactions can have a dramatic effect on estimation precision-even infinitesimally small uncorrelated dephasing noise, modelled as a semigroup (time-homogeneous-Lindbladian) evolution [10], forces the MSE to eventually follow the SQL [11]. This result was proven to be an instance of the quantum Cramér-Rao bound (QCRB) [12] for generic Lindbladian dephasing and thus holds even when using optimized entangled states and measurements [13][14][15][16]. The question then arises of what is the ultimate precision limit when the noisy time evolution is not governed by a dephasing dynamical semigroup [13][14][15][16][17][18][19][20][21][22][23][24][25][26]. The SQL has been shown to be surpassable in the presence of time-inhomogeneous (non-semigroup) dephasing noise [24], noise with a particular geometry [25] and correlated timehomogeneous dephasing [27], or when the noise geometry allows for error correction techniques [28].Here, we derive the ultimate lower bounds on the MSE for the noisy frequency estimation scenario depicted in Fig. 1 where probe systems are independently affected by the decoherence. In particular, we focus on uncorrelated phase-covariant noise, that is, noise-types commuting with the parameter-encoding Hamiltonian, as these underpin the asymptotic SQL-like precision in the semigroup case [16,25]. Yet, most importantl...

We establish general limits on how precise a parameter, e.g., frequency or the strength of a magnetic field, can be estimated with the aid of full and fast quantum control. We consider uncorrelated noisy evolutions of N qubits and show that fast control allows to fully restore the Heisenberg scaling (∼ 1/N 2 ) for all rank-one Pauli noise except dephasing. For all other types of noise the asymptotic quantum enhancement is unavoidably limited to a constant-factor improvement over the standard quantum limit (∼ 1/N ) even when allowing for the full power of fast control. The latter holds both in the single-shot and infinitelymany repetitions scenarios. However, even in this case allowing for fast quantum control helps to improve the asymptotic constant factor. Furthermore, for frequency estimation with finite resource we show how a parallel scheme utilizing any fixed number of entangled qubits but no fast quantum control can be outperformed by a simple, easily implementable, sequential scheme which only requires entanglement between one sensing and one auxiliary qubit.

We study how useful random states are for quantum metrology, i.e., whether they surpass the classical limits imposed on precision in the canonical phase sensing scenario. First, we prove that random pure states drawn from the Hilbert space of distinguishable particles typically do not lead to superclassical scaling of precision even when allowing for local unitary optimization. Conversely, we show that random pure states from the symmetric subspace typically achieve the optimal Heisenberg scaling without the need for local unitary optimization. Surprisingly, the Heisenberg scaling is observed for random isospectral states of arbitrarily low purity and preserved under loss of a fixed number of particles. Moreover, we prove that for pure states, a standard photon-counting interferometric measurement suffices to typically achieve resolution following the Heisenberg scaling for all values of the phase at the same time. Finally, we demonstrate that metrologically useful states can be prepared with short random optical circuits generated from three types of beam splitters and a single nonlinear (Kerr-like) transformation.

We find the optimal scheme for quantum phase estimation in the presence of loss when no a priori knowledge on the estimated phase is available. We prove analytically an explicit lower bound on estimation uncertainty, which shows that, as a function of number of probes, quantum precision enhancement amounts at most to a constant factor improvement over classical strategies.